You need to know about convergence, and most likely natural logarithms also. (Convergence of an infinite product is often equated to convergence of the series of natural logs of (all but finitely many of) the factors, assuming a branch of the log for which ln(1) is 0.

So probably first semester calculus at earliest -- late HS or early college.

You might not run into infinite products until you take complex analysis. That would be a bit later. An amusing example is

which converges for |z| < 1. Premultiply by 1 - z and watch what happens